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Commit 7bd81231 authored by Luke Naylor's avatar Luke Naylor
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Add plot for characteristic curves in rank 0 case

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......@@ -282,9 +282,7 @@ Considering the stability conditions with two parameters $\alpha, \beta$ on
Picard rank 1 surfaces.
We can draw 2 characteristic curves for any given Chern character $v$ with
$\Delta(v) \geq 0$ and positive rank.
These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$, and
are illustrated in Fig \ref{fig:charact_curves_vis}
(dotted line for $i=1$, solid for $i=2$).
These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$.
\begin{definition}[Characteristic Curves $V_v$ and $\Theta_v$]
Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
......@@ -296,9 +294,21 @@ define two characteristic curves on the $(\alpha, \beta)$-plane:
\end{align*}
\end{definition}
\begin{fact}[Geometry of Characteristic Curves]
\subsection{Geometry of the Characteristic Curves}
These characteristic curves for a Chern character $v$ with $\Delta(v)\geq0$ are
not affected by flipping the sign of $v$ so it's only necessary to consider
non-negative rank.
As discussed in subsection \ref{subsect:relevance-of-V_v}, making this choice
has Gieseker stable coherent sheaves appearing in the heart of the stability
condition $\firsttilt{\beta}$ as we move `left' (decreasing $\beta$).
\subsubsection{Positive Rank Case}
\label{subsect:positive-rank-case-charact-curves}
\begin{fact}[Geometry of Characteristic Curves in Positive Rank Case]
The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
as well as the restrictions on $v$:
as well as the restrictions on $v$, when $\chern_0(v)>0$:
\begin{itemize}
\item $V_v$ is a vertical line at $\beta=\mu(v)$
\item $\Theta_v$ is a hyperbola with assymptotes angled at $\pm 45^\circ$
......@@ -312,21 +322,8 @@ as well as the restrictions on $v$:
\end{itemize}
\end{fact}
\subsection{Relevance of the Vertical Line $V_v$}
By definition of the first tilt $\firsttilt\beta$, objects of Chern character
$v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
$\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
fact, if there is a Gieseker semistable coherent sheaf $E$ of Chern character
$v$, then $E \in \firsttilt\beta$ if and only if $\beta<\mu(E)$ (left of the
$V_v$), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
Because of this, when using these characteristic curves, only positive ranks are
considered, as negative rank objects are implicitly considered on the right hand
side of $V_v$.
These are illustrated in Fig \ref{fig:charact_curves_vis}
(dotted line for $i=1$, solid for $i=2$).
\begin{sagesilent}
from characteristic_curves import \
......@@ -334,6 +331,7 @@ typical_characteristic_curves, \
degenerate_characteristic_curves
\end{sagesilent}
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
......@@ -358,8 +356,49 @@ degenerate_characteristic_curves
\label{fig:charact_curves_vis}
\end{figure}
\begin{sagesilent}
from rank_zero_case import Theta_v_plot
\end{sagesilent}
\subsubsection{Rank Zero Case}
\begin{fact}[Geometry of Characteristic Curves in Rank 0 Case]
The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
as well as the restrictions on $v$, when $\chern_0(v)=0$ and $\chern_1(v)>0$:
\begin{minipage}{0.49\textwidth}
\begin{itemize}
\item $V_v = \emptyset$
\item $\Theta_v$ is a vertical line at $\beta=\frac{D}{C}$
where $v=\left(0,C\ell,D\ell^2\right)$
\end{itemize}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\sageplot[width=\textwidth]{Theta_v_plot}
%\caption{$\Delta(v)>0$}
%\label{fig:charact_curves_rank0}
\end{minipage}
\end{fact}
\subsection{Relevance of the Hyperbola $\Theta_v$}
\subsection{Relevance of $V_v$}
\label{subsect:relevance-of-V_v}
By definition of the first tilt $\firsttilt\beta$, objects of Chern character
$v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
$\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
fact, if there is a Gieseker semistable coherent sheaf $E$ of Chern character
$v$, then $E \in \firsttilt\beta$ if and only if $\beta<\mu(E)$ (left of the
$V_v$), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
Because of this, when using these characteristic curves, only positive ranks are
considered, as negative rank objects are implicitly considered on the right hand
side of $V_v$.
\subsection{Relevance of $\Theta_v$}
Since $\chern_2^{\alpha, \beta}$ is the numerator of the tilt slope
$\nu_{\alpha, \beta}$. The curve $\Theta_v$, where this is 0, firstly divides the
......@@ -377,6 +416,7 @@ $\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on
$\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0)$
(as per subsection \ref{subsect:bertrams-nested-walls}).
\subsection{Bertram's Nested Wall Theorem}
\label{subsect:bertrams-nested-walls}
......
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